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On the solutions to complex parameter-dependent LMIs involved in the stability analysis of 2D discrete models
Olivier Bachelier, Didier Henrion, Nima Yeganefar, Driss Mehdi
To cite this version:
Olivier Bachelier, Didier Henrion, Nima Yeganefar, Driss Mehdi. On the solutions to complex parameter-dependent LMIs involved in the stability analysis of 2D discrete models. 2014. �hal- 01064115v2�
On the solutions to complex parameter-dependent LMIs involved in the stability analysis
of 2D discrete models
Olivier Bachelier
1, Didier Henrion
2,3,4, Nima Yeganefar
1, Driss Mehdi
1Version of June 2, 2015
Abstract
The aim of this short communiqu´e is to adapt a result established by Bliman, related to the possible approximation of the solutions to real-parameter-dependent linear matrix inequalities (LMIs), to the special context of stability analysis of 2D discrete Roesser models. While Bliman considered the case of LMIs involving several realparameters, which is especially crucial for the analysis of linear systems against parametric deflections, the stability of Roesser models leads to consider LMIs with only one single complexparameter. Extending the results from real parameters to complex ones is not straightforward in our opinion. This is why the present note discusses precautions to be taken concerning this case before applying the results in a 2D context. Actually, it is shown that a well-known condition for structural stability of a 2D discrete Roesser can be relaxed into an LMI system whose solution polynomially depends on a single complex parameter over the unit circle.
1 Motivation and introduction
Multidimensional models raised a growing interest within the community of automatic control during the last three decades, due to the wide field of their applications (signal and image processing, seismic phenomena, ... see [16, 12]). All systems for which information propagates in more than one direction are eligible to be described by such models. Among them, the 2D models are probably the most studied ones, since they can be used to describe the behaviours of, e.g. Iterative Learning Control (ILC)-schemes or so-called
1University of Poitiers, LIAS-ENSIP, Bˆatiment B25, 2 rue Pierre Brousse, TSA 41105, 86073 Poitiers cedex, France.
2CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France.
3Universit´e de Toulouse; F-31400 Toulouse; France.
4Faculty of Electrical Engineering, Czech Technical University in Prague, Technick´a 2, CZ-166 26 Prague, Czech Republic.
repetitive processes [12, 20]. In the present contribution, we focus on 2D Roesser models.
These models comply with
q1x1(j1, j2) q2x2(j1, j2)
=
A11 A12 A21 A22
x1(j1, j2) x2(j1, j2)
, (1)
where xi(j1, j2) ∈ IRki is the state subvector associated with dimension i. qi is either a derivative operator or a shift operator along the ith dimension. Therefore, (1) can be continuous, discrete or mixed. The state vector is divided into two subvectors correspond- ing to the propagation of the information in two directions. The matrices Aij are here assumed to be real.
Besides, we define the following subsets of the extended complex plane lC = lC ∪ {∞},
Di = {λ∈ lC :fi(Ri, λ)<0}, DCi = {λ∈ lC :fi(Ri, λ)≥0},
∂Di = {λ∈ lC :fi(Ri, λ) =fi( ˆRi, λ) = 0},
D⋄i = {δ∈ lC :δ−1 ∈ DiC}={δ∈ lC :fi( ˆRi, δ)≥0}
(2)
with
fi(Ri, λ) = λ
1 ′
Ri λ
1
, Ri =
ri11 ri10 ri10 ri00
∈ IR2×2, Rˆi =
ri00 ri10 ri10 ri11
. (3)
The matricesRi are restricted to cases where eitherRi =
0 1 1 0
orRi =
1 0 0 −1
. In such cases Di corresponds to the usual stability region, i.e. to the open left half complex plane or the open unit disc respectively. DiC is the complementary region, ∂Di ⊂ DCi is the boundary of Di and D⋄i is a set described by λ−1 when λ describes DCi .
In order to make sense, to each operator qi we associate a region Di as follows:
qixi(j1, j2) = ∂
∂jixi(j1, j2) ⇒ Di = lC−, i= 1,2, q1x1(j1, j2) = x1(j1+ 1, j2) ⇒ D1 = D,
q2x2(j1, j2) = x1(j1, j2+ 1) ⇒ D2 = D,
(4)
where lC−denotes the achieved open left half complex plane andDis the unit disc. Hence, the continuous, discrete and mixed cases can all be taken into account in this framework.
A frequently used stability criterion is as follows.
Lemma 1 System (1) is stable if and only if A22 is D2-stable andM(δ) =A11+A12(I− δA22)−1δA21 is D1-stable for any δ∈∂Di (or any δ∈ Di⋄.
The justification can be found for example in [15] for the countinous case, and in [24] for the other cases. From this let us derive the next lemma.
Lemma 2 System (1) is stable if and only if there exist a positive definite matrix Y and a positive definite parameter-dependent Hermitian matrix P(δ) such that
r200Y +r210A′22Y +r210Y A22+r211A′22Y A22<0 (5) and ∀δ∈∂D2(resp.∀δ∈ D2⋄),
r100P(δ) +r110M′(δ)P(δ) +r110P(δ)M(δ) +r111M′(δ)P(δ)M(δ)<0. (6) Proof. This is just an interpretation of Lemma 1 in terms of Lyapunov’s or Stein’s inequalities.
Lemma 2 is interesting but involves condition (6) which is not computationally tractable due to its dependence on the complex parameterδ. However, if one could prove thatP(δ) might comply to a specified non conservative structure, it could allow the research for tractable relaxations. The establishment of such relaxations is the underlying motivation behind this work. But the present note focuses on the preliminary step i.e. the relevance of the structure that can be assumed forP(δ). In the sequel we discuss this issue in order to prove the main result which is now stated.
Theorem 1 System (1) is stable if there exist positive definite matrices Y and
P(δ) = Xν
i=1
Piδi
!H
, Pi ∈ lCn×n, (7)
(whereXH denotes X+X′) such that (5) and (6) are satisfied. Moreover, the condition is also necessary for a sufficiently large value of the degree ν.
Theorem 1 was used in the discrete case in [1] to derive a simple and rather tractable hierarchy of sufficient LMI conditions for (5-7), which tends to necessity, by using a clas- sical S-procedure as formulated in [23]. Unfortunately, in [1], Theorem 1 was taken at a face value with no actual proof. More precisely, the result in [1] was based upon [5] but we realized that it was not technically correct. This paper aims at filling this gap. To fully motivate Theorem 1, we have to explain the interest in applying the S-procedure whereas there exist more sophisticated relaxation schemes such as moment-SOS (sum of squares), or Lasserre’s hierarchy [17]. Indeed, it could be possible to write δ = a+ib with (a;b) ∈ IR2 and i being the imaginary unit, and then to prove the existence of a solution to (6) that would be polynomial with respecto to a and b (see for instance [22]).
Such a polynomial could be computed by using SOS techniques. But the interest in using the S-procedure associated with the Linear Fractional Representation (LFR) framework to eliminate δ ∈ lC from the condition is that this relaxation not only possibly leads to non conservative LMIs in the problem that we aim at tackling, but, moreover, straight- forwardly yields clear expressions of these LMIs [1]. The derivation of these expressions is a key step in the design of stabilizing control laws as highlighted in [3] where a necessary and sufficient LMI condition for state feedback stabilization is obtained... provided that Theorem 1 is proved! Hence our motivation to clarify this technical aspect.
The LMIs are a tool whose utility and popularity are no longer to be demonstrated [6].
They consitute an effective approach to solve numerous problems in automatic control.
Some of those problems unfortunaltely involve parameter-dependent LMIs. Typically, analyzing the stability of linear systems against parameter uncertainties amounts to as- sessing the existence of a solution (i.e. a Lyapunov function) which itself depends on these parameters [11, 13, 8], which is a difficult problem. In order to efficiently relax such a problem, the question of the way this solution could depend on the parameters is of course important. Bliman made a very important step in the understanding of such a question. Indeed, in [5], he proved that when the LMI depends on real parameters, its solution can be considered polynomial with respect to the parameters without conser- vatism. The question of its minimal degree is still open. In the remainder of the paper, inspired by the work of Bliman [5] and motivated by our will to fully justify the work in [1], we discuss the possibility for the solutions to complex parameter-dependent LMIs to be assumed polynomial with respect to the complex parameters. The purpose is not to provide a complete complex counterpart with a complete proof, which would be very redundant with Bliman’s work, but to insist on the steps where some adaptations are required and to highlight the currently existing limits which prevent the complete gen- eralization of Bliman’s result. But it must be kept in mind that the case with only one complex parameter is of special interest for us to prove Theorem 1.
2 Bliman’s result
In this section, we recall Bliman’s theorem and give a very brief outline of his proof. Let us define the following expression:
G(x, δ) =G0(δ) +
p
X
i=1
xiGi(δ), δ∈K, x=
x1
... xp
∈ IRp, (8)
where K is a compact subset of IRm and G0, G1, ..., Gp are mappings defined inK and taking values in the set of symmetric matrices of IRn×n.
Theorem 2 [5] Assume that mappings Gi, i = 1, ..., p, are continuous. If for any δ∈K, there exists x(δ) such that G(x(δ), δ)>0, then there exists a polynomial function x∗ :K → IRp such that for any δ∈K, G(x∗(δ), δ)>0.
The result is very strong for two reasons. First, it proves that there exists a functionx∗(δ) which can be used as a solution for every δ. Moreover, it states that this function can be polynomial. It is not worth detailing the proof which can be found in [5] but it is useful, for the sequel, to recall an outline.
Outline of the proof of Theorem 2. The proof consists in four steps:
i) It starts with noting that, if for anyδ∈K, there exists x(δ) such thatG(x(δ), δ)>0 holds, then there existsα >0 such that for anyδ∈K, the inequalityG(x, δ)≥2αI has a nonempty set of solutions. It leads to define the set-valued map F which, for any δ∈K, associates the nonempty closed convex set
F(δ) = {x∈ IRp :G(x, δ)≥αI}. (9)
(Note that anyx∈F(δ) satisfies G(x, δ)>0.) ii) Then the lower semicontinuity of F is proved.
iii) Michael’s selection theorem [19] is invoked to prove that, from F, there exists a continuous selection f mapping elements of K into elements of IRp and such that, for any δ∈K, G(f(δ), δ)≥ αI.
iv) The last step applies Weierstrass approximation theorem [9] to prove that each real- valued entry off is the limit of a sequence of polynomials, in the sense of uniform convergence.
Our purpose leads us to discuss the extension of Theorem 2 where K is a compact subset of lCm and where each Gi(δ) belongs to lCn×n. The three first steps of Bliman’s proof can be preserved. All the discussion concerns the fourth step. We will show that the extension to the complex case is not so obvious and deserves a little attention.
3 The complex case
In this section, we discuss to what extend complex parameters can be considered in The- orem 2. But before doing so, we remind the reader of the needed mathematical theorems.
Theorem 3 (Weierstrass approximation theorem [21]) Suppose that f is a continuous real-valued function defined on the real range [a, b]. Then there exists a sequence of real- valued polynomial functions which uniformly converges to f.
In other words, it is possible to approximate a continuous real-valued function f by a real-valued polynomial function with any desired accuracy, provided that the degree of the underlying polynomial is large anough. Bliman actually applies an extension which is the Stone-Weierstrass Theorem.
Theorem 4 (Stone-Weierstrass theorem [25, 9, 21]) Let K be a compact Hausdorff space and let P be a subalgebra of the set C(K, IR ) of continuous functions from K to IR, then P is dense in C(K, IR ) if and only if it includes a non zero constant function and separates the points.
Indeed, a subset K ∈ IRm is a compact Hausdorff space and the set of polynomial functions with arguments δ taken in K is a subalgebra of C(K, IR ). It would also be possible to consider K as a subset of lCm. However, we also want to consider matrices G(δ) in lCn×n and thereforeC(K, lC ), the set of continuous functions fromK to lC , should be involved in Theorem 4 instead of C(K, IR ). But such a substitution in the writing of the theorem is not possible. The extension is not so simple but it exists and it is due to Glimm [14]. We here propose a simplified version which is especially taylored for our needs.
Theorem 5 (Complex version of Stone-Weierstrass theorem [14]) Let K be a compact Hausdorff space and letC(K, lC )be the set of continuous functions from K to lC. Also let P be a subalgebra of C(K, lC ) inluding a non zero constant function and which separates the points. ThenP is dense in C(K, lC )if and only if it is a C*-algebra with ’*’ being the relation which associates two complex numbers by conjugation.
This theorem allows the consideration of complex functions but also adds an additional property to be checked byP. This is a key issue in our motivation to write this paper since the set of polynomial functions fromK to lC is not stable under conjugation. Therefore, we have to carefully choose the set P to make this theorem valid.
Before stating our result, we also recall the next theorem, which is another avatar of Weierstrass approximation theorem.
Theorem 6 (Mergelyan’s theorem [18, 21]) Let K be a compact subset of lC such that lC\K is connected. Any continuous function f : K → lC which is holomorphic on its interior int(K) can be uniformly approximated on K by a polynomial function.
Bliman’s theorem (Theorem 2) relies on the two first theorems recalled in this section.
With the two other ones, we are now able to state the next theorem which is a key result before proving Theorem 1.
Theorem 7 Let the following expression be defined:
G(x, δ) =G0(δ) +
p
X
i=1
xiGi(δ), δ∈K, x=
x1
. . . xp
∈Clp, (10)
where K is a compact subset of lCm andG0, G1, ...,Gp are continuous mappings defined in K and taking values in the set of Hermitian matrices of lCn×n. Also define K′ ⊆ lCm as the image of K under conjugation. If for any δ ∈ K, there exists x(δ) such that G(x(δ), δ) > 0, then there exists x∗ : K×K′ → lCp, a polynomial function with respect to δ and δ′ such that, for any δ ∈ K, G(x∗(δ, δ′), δ) > 0. Moreover, if m = 1, if lC\K is connected, and if int(K) = ∅, there exists x• : K → lCp, a polynomial function with respect to δ, such that, for any δ in K, G(x•(δ), δ)>0.
Proof. Once again, we remind the reader that the three first steps of the proof of Theo- rem 2 [5] can be reproduced mutatis mutandis. Only the fourth step deserves attention.
Once we know that there existspcontinuous functions fromK to lC which are the entries of a solutionx(δ) such that G(x(δ), δ)≥αI for someα >0, it remains to prove that, for any given complex entry off, there exists an approximation.
Define the set P of functions ofδ defined through the expressions
f(δ) = Xµ
i=0
Xµ
j=0
cijδiδ′j, cij ∈ lC. (11) These are polynomial functions w.r.t. both δ and δ′. It is quite easy to check that P is a subalgebra of C(K, lC ) which separates the points and contains non zero constant
function. But in addition, P is closed under conjugation so it is a C*-algebra. Therefore, Theorem 5 can be invoked to claim that each entry xi(δ), i= 1, ..., p, of x(δ) is the limit of a sequence of polynomial functions complying with (11). Letx∗(δ, δ′) denote (for short x∗(δ)) the lCp-valued function resulting from the concatenation of these limits. Then it comes
∀δ∈K, G(x∗(δ), δ)≥αI ⇒ ∀δ ∈K, G(x∗(δ), δ)>0. (12) This proves the first part of the theorem.
To prove the remainder, it can be noticed that if int(K) =∅, the elements of P do not need to be holomorphic on K but just continuous to invoke Theorem 6. Therefore, each entry x∗i(δ), i = 1, ..., p of x∗(δ) can be uniformly approximated on K by a polynomial function
x•i(δ) = Xν
i=0
biδi, bi ∈ lC. (13)
Let denotex•(δ) the lCp-valued function resulting from the concatenation of the functions x•i(δ). It follows
∀δ∈K, G(x•(δ), δ)>0. (14)
The previous theorem deserves several comments. The first statement in Theorem 7 is what will really be exploited in the sequel. It shows that the solution to complex parameter-dependent LMIs can be assumed polynomial with respect to both the param- eter vector δ and its conjugate δ′, with no loss of generality. This is of course a very in- teresting consequence of Glimm’s theorem but the research for computationally tractable relaxations of the original LMI conditions often suggests that the solutions should rather be polynomial or rational with the entries of δ (not those of δ′, see [1, 2, 3]). This ad- ditional step on the way to useful approximations is not so easy to make. Mathematical literature dedicated to approximations usually evokes Mergelyan’s theorem (Theorem 6) as the most significant and advanced step but it can be seen that it is restricted toδ∈ lC . Furthermore,δ should belong to a subset which should satisfy some special properties and these properties are not encountered in many practical problems, including the present one. For this reason, we claim that the research for efficient approximations of solutions to complex LMIs with one or several complex parameters, through simple functions, is a widely open problem. More precisely, consider G(x, δ) expressed as in (8) or (10) and the underlying functiong(x, δ, y) =y′G(x, δ)y which should be non negative: If there are various tools to numerically approximate solutions x(δ, y) when δ ∈ IRm and y ∈ IRn, the existence of such approximations when δ ∈ lCm and y ∈ lCn is not as obvious as in the real case. Nevertheless, the very special case addressed in the present contribution offers a possibility to circumvent the obstacle.
If however one wants to enlarge the possibilities of application of Theorem 5 to m ≥ 1, the following corollary might be interesting.
Corollary 1 Let an expression be defined by (10) where K is a compact subset of lCm and G0, G1, ..., Gp are continuous mappings defined in K and taking values in the set of Hermitian matrices of lCn×n. Also define K′ ⊆ lCm as the image of K under conjugation.
If for anyδ∈K, there existsx(δ)such thatG(x(δ), δ)>0, then there existsx∗ :K×K′ → lCp, a rational function with respect to δ =
δ1′ . . . δm′ ′
and δ′, which complies with implicit LFR (ILFR)-based form
x∗(δ) =D∗+C∗(E∗−∆A∗)−1(∆B∗−F∗), (15) where A∗, B∗, C∗, D∗, E∗ and F∗ are either real or complex matrices of appropriate dimensions, where ∆ = ∆⊕∆′ (⊕ denoting “blocdiag”), where ∆ =⊕mi=1δiIqi, and where qi ≥0, i= 1, ..., m, such that, for any δ∈K, G(x∗(δ, δ′), δ)>0.
Proof. (Outline) The proof is more or less the same as the first part of the proof of The- orem 7 by noting that when associating two ILFRs of the form (15) (by multiplication or addition), ones gets another ILFR w.r.t. ∆1⊕∆2where ∆j = ∆j⊕∆′j and ∆j =⊕mi=1δiIqji (note that the only difference between ∆1 and ∆2 are the dimensions q1i and q2i). It is well known that the rows and and the columns of the matrices involved in the ILFR can be re-arranged while preserving the expression and so as it becomes an ILFR w.r.t. ∆ as defined in the corollary. The consequence is that the set of these ILFRs has clearly the properties of a subalgebra. By applying the conjugation on such an ILFR, another ILFR is obtained, w.r.t. ∆′ = ∆′⊕∆. By again invoking the possible re-arrangement on the rows and the columns of the involved matrices, this ILFR can be written as an ILFR w.r.t. to ∆. Therefore, the set is a C∗-algebra and Theorem 5 can be applied.
This corollary will be applied in the next section
4 Stability of the nD Roesser models
Now, we come back to our main motivation and prove Theorem 1. We also propose other perspectives induced by Corollary 1.
4.1 2D Roesser models
Proof of Theorem 1. Sufficiency holds by virtue of Lemma 2. The interest lies in the second statement and in the assumed structure of P(δ). First note that if P(δ) is solution, so isP′(δ) =P(δ) and thusPH(δ). Therefore, there exists Q(δ) (not necessarily Hermitian) such that,
∀δ∈∂D2(or ∀δ∈ D⋄2),
M(δ) =r100QH(δ) + (r110M′(δ)QH(δ))H +r111M′(δ)QH(δ)M(δ)<0
QH(δ)>0 (16)
Such a system is an LMI system parametered byδ. Following the notations of the previous sections, it can be rewritten asG(x(δ), δ)>0 where G(δ) is deduced fromM(δ) and x(δ)
contains the decision variablesi.e. the entries of Q(δ) [6]. The parameter δ describes the unit circle or the extended imaginary axis which is a compact set K. Matrix M(δ) is rational w.r.t. δ and, since A22 is stable, M(δ) is continuous in K (it has no pole inside K). So matrices Gi(δ) are also continuous. Therefore, by virtue of Theorem 7, Q(δ) can be chosen polynomial w.r.t. δ and δ′ without loss of generality provided that the degree of the polynomial is large enough. Q(δ) can then be written
Q(δ) = Xη
k=1
Xγ
l=1
Qklδkδ′l, Qkl ∈ lCn×n. (17) If∂D2 is the unit circle then δδ′ = 1, so each monomial of the form δkδ′l equals either δi orδ′i with i= max(k, l)−min(k, l). Therefore Q(δ) complies with
Q(δ) = Xν
i=1
Xiδi+Yiδ′i, (Xi, Yj)∈ {lCn×n}2. (18) Since P(δ) =QH(δ), it can be written as in (7) with Pi =Xi+Yi′.
If∂D2 is the imaginary axis, then δ′ =−δ so (17) obviously leads to (7). Then the proof is completed for ∂D2, not for D⋄2. For that, assume that P(δ) has the required form (7) and is solution for δ∈∂D2. Also consider the functions fi(δ) defined by
fi(δ) =λi−1(P(δ)⊕(−M(δ))), i= 1, . . . ,2k1, (19) over D2⋄, where λi(.) are the eigenvalue of the matrix argument (which are real since the argument is Hermitian). Since P(δ)⊕(−M(δ)) is Hermitian, each eigenvalue is analytic w.r.t. δ. By invoking the maximum modulus principle [21], the functionsfi(δ) reach their maximum modulus over∂D2. Since from the first part of the proof,P(δ)⊕(−M(δ)) does not become singular over ∂D2, the various fi(δ) are bounded over∂D2. Thus, they area fortioribounded overD⋄2 (they cannot tend to infinity). Therefore P(δ)⊕(−M(δ)) never becomes singular over D⋄2. Consequently, P(δ) remains positive definite, M(δ) remains negative definite and thus P(δ)>0 remains a solution to (6) over D2⋄.
Remark 1 When D2 is the open left half complex plane, it is explained in [2] that another admissible expression of P(δ) (which could be justified through quite similar arguments) is
PG(δ) = Xα
h=0
Qh δ
1 +δ h!H
, Qh ∈ lCk1×k1. (20) Indeed, following the discussion of section 1, the LFRs involved in the S-procedure used to obtain non parameter-dependent LMIs would not be well-posed with (7) whereas thery are with (20).
It must be noticed that this result generalizes the result obtained, through a different approach, in [4] for the discrete case. Here, all the cases are embedded in the same framework.
4.2 Insights into n D Roesser models
Consider a generalization of model (1) to dimension n:
q1x1(j1, ..., jn) ...
qnx2(j1, n..., jn)
=
A11 . . . A12 ... . .. ... An1 . . . Ann
x1(j1, ..., jn) ... xn(j1, . . . , jn)
, (21) where qi denotes either shift operator or derivative along dimension i. In many appli- cations of nD models, one dimension (most probably the time) is not considered on an equal footing and one is mainly interest in the stability w.r.t. that dimension. Let us consider with no loss of generality that this dimension is the first one. Then consider the transform X : xi(j1, j2, , ..., jn) 7→ Xi(j1, λ2, ..., λn) where X actually corresponds to the Laplace transform for the continuous dimensions and the Z transform on the discrete di- mension. If we define a “mixed”L2−l2-norm ofXi, denoted by ||Xi||2, then the dynamics of x1 is related to the behaviour of||X1||2 under the constraint
q1X1(j1, λ2, ..., λn) =DM +CM
I−∆˜AM−1
∆˜BM
| {z }
M(δ)
X1(j1, λ2, ..., λn), (22)
where j1 is a continuous or discrete time, λi, i = 2, ..., n describes either the unit circle or the imaginary axis, whereδ=
(δ1 = λ1
2)′ . . . (δn−1 = λ1
n)′ ′
, where ˜∆ =⊕ni=1δn−1, and where
AM BM CM DM
=
A22 . . . A2n A21 ... . .. ... ... An1 . . . Ann An1 A12 . . . A1n A11
. (23)
Thus, in the sense of ||.||2, the system is asymptotically stable along dimension 1 if and only ifM(δ) has no eigenvalue outside ofD1(i.e. outside of lC−orD) for anyδi describing either the unit circle or the imaginary axis, depending on the nature of the propagation along dimension i+ 1. This straightforwardly results in the next theorem.
Theorem 8 System (21) is time-relevant stable in the sense of ||.||2 if and only if In- equality (6) holds for any δ ∈ ∂D2 ×...×∂Dn, where the various ∂Di are defined by (2) and by generalizing the constraint (4) to dimension n.
Therefore we state the next theorem
Theorem 9 If system (21) is time-relevant stable in the sense of ||.||2, then there exists a solution P•(δ)to Inequality (6) over ∂D2×...×∂Dn which complies with the ILFR-based expression
P•(δ) = D•+C•(E•−∆Aˆ •)−1( ˆ∆B•−F•), (24) where ∆ =ˆ ⊕mi=1δiIgi and gi ≥0, i= 1, ...,(m=n−1).
Proof. From Corollary 1, it can be easily assessed that there exists a solution P∗(δ) = D∗ +C∗(E∗−∆A∗)−1(∆B∗ −F∗) with ∆ = ∆⊗∆′ ∆ = ⊕mi=1δiIqi, and qi ≥ 0. Since δi belongs to either the imaginary axis or the unit circle, it can be written −δi or δ−i 1. Those two expressions are also special instances of ILFRs. Since ILFRs of ILFRs are also ILFRS, then it is possible to rewrite the solution P∗ asP• given in (24).
The question is now that of (possibly exactly) relaxing conditions such as (6). It cannot be addressed here unless by unduly increasing the size of the article. However, whenn = 2 (i.e. m = 1), Theorem 1 was exactly relaxed in [1] in the purely discrete case through S-prcocedure. Using a close by different approach, the mixed case was efficiently tackled in [7] where SOS-relaxations were prefered to S-procedure. Moreover, in [7], upperbounds on the degree of the polynomial are provided, which is very interesting from a theoretical point of view. But the bounds are still far larger than the degrees actually required to assess stability, and therefore might not be easily exploited on practical examples.
An exact LMI-based relaxation is provided in [10]. It is very interesting because unlike the other approaches, it does not require any degree to be guessed. Therefore, instability can be proved by such a condition. Nevertheless, it is restricted to the discrete case, and considers Fornasini-Marchesini (FM) models instead of Roesser (R) models. Of course, FM and R models can be converted to each other in the control-free case but it is no longer true in the presence of a control input. So any attempt to extend the result to design may clearly separate the two models. Moreover, examples exposed in [2] tend to show that the condition of [10], unless more direct, may not necessarily be more attractive than [7, 2] in terms of computation time. Furthermore, we would like to insist that [2]
exploits Theorem 1 (or more precisely a slightly modified version) to encompass all the cases (discrete, continuous, and mixed).
At last, in [3], the first exploitation of Theorem 1 is made for control and a nesssary and sufficient condition for state feedback structural stabilization of 2D discrete models is proposed. This is a major advantage compared to other approaches. The generalization to continuous and mixed cases is under investigation.
Forn >2 (orm >1), the S-procedure-based reasoning used in [2] might be extended but also probably lead to a conservative LMI condition, hence our focus on the 2D case.
Once again, the reader must be aware that the previously evoked results [1, 2, 3] rely on Theorem 1 whereas it had not been clearly proved yet. The main contribution of the present paper is to fill this gap and to embed this result into a more general formalism.
5 Conclusion
In this note, we proposed a necessary and sufficient condition for the stability of 2D (discrete, continuous or mixed) Roesser models which consists of an LMI system where the solutions are polynomial w.r.t. a complex parameter describing either the unit circle or the imaginary axis (or either the closed unit disc or the closed right half-plane). Insights to generalize the results to higher dimensions were also proposed. It has to be noticed that this result is in accordance with what was obtained in the discrete case through a
completely different approach in [4] and that a recent paper proposed a quite similar result in the mixed continuous-discrete-case [7]. But the present approach can offer a general method to derive exact relaxations of 2D-stability conditions for all the cases [2]. At last, but not least, a first exploitation of this result for the synthesis of control laws was made in [3]. Therefore, Theorem 1 is a keystone of a promising approach.
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